Finite nonsolvable groups with many distinct character degrees

Let G be a finite group and let Irr(G) denote the set of all complex irreducible characters of G. Let cd(G) be the set of all character degrees of G. For a degree d ∈ cd(G), the multiplicity of d in G, denoted by m_G(d), is the number of irreducible characters of G having degree d. A finite group G is said to be a T_k-group for some integer k≥1 if there exists a nontrivial degree d0 ∈ cd(G) such that m_G(d₀) = k and that for every d ∈ cd(G)-{1, d₀}, the multiplicity of d in G is trivial, that is, mG(d) D 1. In this paper, we show that if G is a nonsolvable T_k-group for some integer k ≥ 1, then k = 2 and G≅PSL₂(5) or PSL₂(7).